A Data Mining-Based Framework for Multi-item Markdown Optimization

Abstract

Markdown decisions in retailing are made based on the demand forecasts which may or may not be accurate in the first place. In this chapter, we propose a framework for forecasting weekly demands of retail items via linear regression models within multi-item groups that incorporate both positive and negative item associations. We then utilize dynamic pricing models to optimize markdown decisions based on the forecasts within multi-item groups. Grouping items can be considered as a form of variable selection to prevent the overfitting in prediction models. We report regression results from multi-item groupings besides results from single-item regression model on a real-world dataset provided by an apparel retailer. We then report markdown optimization results for the single items and multi-item groupings that multi-item forecasting models are built upon. The results show that the regression models provide better estimates within multi-item groups compared to the single-item model. Moreover, the overall revenues achieved in multi-item markdown optimization across all grouping schemes are higher than the total revenue yielded by single-item markdown optimization scheme.

Appendices

Appendix

Multi-item Markdown Optimization Model

The parameters of the model are given below:  

n :

number of products

$n_l$ :

number of products in cluster l

T :

number of weeks in the season

$T_l$ :

planning horizon (weeks) for cluster l

$\varepsilon$ :

very small price value (such as 1 cent), which will allow to differentiate between two prices

${\textit{IS}}_i$ :

initial inventory for product i (at the beginning of the season)

${\textit{IP}}_i$ :

initial price of product i (at the beginning of the season)

$\beta _{0}^{i}, \ldots , \beta _{n_l}^{i}$ :

coefficients for the products in cluster l that represent how they contribute to the sales of product i

$h_{it}$ :

unit cost of holding product i in inventory during week t

${\textit{sV}}_i$ :

salvage value/price of product i (can be fixed as parameters, if desired)

${\textit{lMS}}$ :

maximum number of markdowns throughout a season (optional, fixed by the retail managers)

M :

a large number

 

The variables of the model are given below:

 

$p_{it}$ :

price of product i during week t

$B_{it}$ :

binary variable that indicates whether a markdown is applied to product i in week t (1 if markdown was applied)

$r_{it}$ :

binary variable that indicates whether the demand forecast for product i in week t is positive (1 if demand forecast is positive)

${\textit{wFD}}_{it}$ :

demand forecast for product i for week t (even though this can be a general function of $p_{it}$ and time, constraint (1) models the special case where it is a linear function of $p_{it}$)

$D_{it}$ :

positive demand forecast for product i for week t

$S_{it}$ :

sales of product i in week t

${\textit{fS}}_i$ :

number of units of product i left in inventory at the end of the season

${\textit{TS}}_i$ :

total sales of product i throughout the season

${\textit{wIS}}_{it}$ :

initial inventory of product i in week t

${\textit{wFS}}_{it}$ :

ending inventory of product i in week t

${\textit{hC}}_{it}$ :

total cost of inventory for product i during week t

${\textit{THC}}_{i}$ :

total cost of inventory for product i throughout the season

${\textit{nMS}}_{i}$ :

number of markdowns applied for product i

 

$$\begin{array}{l} {\mathop {\max } \displaystyle \sum _{i=1}^{n_l} \displaystyle \sum _{t=1}^{T_l}p_{it} S_{it} +\displaystyle \sum _{i=1}^{n_l}sV_{i} {{\textit{fS}}}_{i} - \displaystyle \sum _{i=1}^{n_l}\sum _{t=1}^{T_l}h_{it} {{\textit{wFS}}}_{it} } \\ {s.t.} \end{array} $$

$$\begin{aligned} {\textit{wFD}}_{it}= & {} \beta _{0}^{i} + \displaystyle \sum _{j=1}^{n_l}\beta _{j}^{i} p_{jt} +\beta _{n_l+1}^{i} t ~~~\forall i \end{aligned}$$

(6)

$$\begin{aligned} {\textit{wFS}}_{it}= & {} {\textit{wIS}}_{it} -S_{it} ~~~ \forall i,\forall t \end{aligned}$$

(7)

$$\begin{aligned} {\textit{wIS}}_{i1}= & {} {\textit{IS}}_{i} ~~~ \forall i \end{aligned}$$

(8)

$$\begin{aligned} {\textit{TS}}_{i}= & {} \sum _{t=1}^{T_l} S_{it} ~~~ \forall i \end{aligned}$$

(9)

$$\begin{aligned} {\textit{fS}}_{i}= & {} {\textit{IS}}_{i} - {\textit{TS}}_{i} ~~~ \forall i \end{aligned}$$

(10)

$$\begin{aligned} {\textit{wIS}}_{it+1}= & {} {\textit{wIS}}_{it} - S_{it} ~~~ \forall i,\forall t \end{aligned}$$

(11)

$$\begin{aligned} D_{it}\le & {} M r_{it} ~~~\forall i,\forall t \end{aligned}$$

(12)

$$\begin{aligned} D_{it} - {\textit{wFD}}_{it}\le & {} M \left( 1-r_{it} \right) ~~~\forall i,\forall t \end{aligned}$$

(13)

$$\begin{aligned} S_{it}\le & {} D_{it} ~~~ \forall i,\forall t \end{aligned}$$

(14)

$$\begin{aligned} {\textit{wIS}}_{it}\ge & {} S_{it} ~~~ \forall i,\forall t \end{aligned}$$

(15)

$$\begin{aligned} p_{it+1}\le & {} p_{it} \quad ~~~\forall i,\forall t \end{aligned}$$

(16)

$$\begin{aligned} p_{iT}\ge & {} {\textit{sV}}_{i} \quad ~~~\forall i \end{aligned}$$

(17)

$$\begin{aligned} p_{i1}\le & {} {\textit{IP}}_{i} \quad ~~~\forall i \end{aligned}$$

(18)

$$\begin{aligned} p_{it+1} -p_{it} +\varepsilon\le & {} M (1-B_{it+1} )~~~ \forall i,\forall t \end{aligned}$$

(19)

$$\begin{aligned} p_{i1} - {\textit{IP}}_{i} +\varepsilon\le & {} M (1-B_{i1} ) ~~~ \forall i \end{aligned}$$

(20)

$$\begin{aligned} p_{it+1} -p_{it} + M B_{it+1}\ge & {} 0~~~ \forall i,\forall t \end{aligned}$$

(21)

$$\begin{aligned} p_{i1} - {\textit{IP}}_{i} + M B_{i1}\ge & {} 0~~~ \forall i \end{aligned}$$

(22)

$$\begin{aligned} {\textit{nMS}}_{i}= & {} \sum _{t=1}^{T_l} B_{it} ~~~ \forall i \end{aligned}$$

(23)

$$\begin{aligned} {\textit{nMS}}_{i}\le & {} {\textit{lMS}}~~~ \forall i \end{aligned}$$

(24)

$$\begin{aligned} {\textit{hC}}_{it}= & {} h_{it} {\textit{wFS}}_{it} ~~~ \forall i,\forall t \end{aligned}$$

(25)

$$\begin{aligned} {\textit{THC}}_{i}= & {} \sum _{t=1}^{T_l} {\textit{hC}}_{it} ~~~ \forall i \end{aligned}$$

(26)

$$\begin{aligned} p_{it}\ge & {} 0, B_{it} \in \{ 0,1\} , r_{it} \in \{ 0,1\} \quad \quad \forall i,\forall t \\ sV_{i}\ge & {} 0,fS_{i} \ge 0, {\textit{TS}}_{i} \ge 0, {\textit{THC}}_{i} \ge 0\quad \quad \forall i \\ D_{it}\ge & {} 0, {\textit{hC}}_{it} \ge 0, {\textit{wFD}}_{it} ~~ \mathrm{unbounded} ~~~ \forall i,\forall t \\ S_{it}\ge & {} 0, {\textit{wFS}}_{it} \ge 0, {\textit{wIS}}_{it} \ge 0\quad \quad \forall i,\forall t \\ {\textit{nMS}}_{i}\ge & {} 0~~~ {\text {and integer}} \quad \quad \forall i \end{aligned}$$